In this notebook, you can play with the design parameters to regenerate different spiral in-out patterns (so, we draw as many spiral arches as the number of shots). You can play with the number of shots by changing the under-sampling factor.
#DISPLAY T2* MR IMAGE
'''
#DISPLAY BRAIN PHANTOM
在医学成像(如 MRI、CT)和信号处理中,幻影(Phantom)是指用于测试、校准或研究的人工数据或物理模型。
它通常是一个合成(模拟)图像或实验装置,用来模拟真实世界的结构,比如大脑、心脏或其他人体组织。
'''
# 1. 导入必要的库
%matplotlib inline
import numpy as np
import os.path as op
import os
import math ; import cmath
import matplotlib.pyplot as plt
import sys
import brainweb_dl as bwdl # BrainWeb MRI 数据下载
import mrinufft # MRI 非均匀 FFT (NUFFT) 处理库
from mrinufft.trajectories import display
from mrinufft import get_density # 计算密度补偿
from mrinufft.density import voronoi
from mrinufft.trajectories import initialize_2D_spiral
from mrinufft.trajectories.display import display_2D_trajectory # MRI 轨迹可视化
from skimage import data, img_as_float, io, filters # 处理 MRI 图像
from modopt.math.metrics import ssim # 计算结构相似度 SSIM
# 2. 设置 Matplotlib 显示参数
plt.rcParams["image.origin"]="lower" # 设置图像的原点在左下角(默认是左上角)。
plt.rcParams["image.cmap"]='Greys_r' # 设置图像颜色映射为灰度反转(黑白)。
# 3. 下载并处理 MRI 图像
mri_img = bwdl.get_mri(4, "T1")[70, ...].astype(np.float32)
'''
bwdl.get_mri(4, "T1") : 从 BrainWeb 数据库下载一个 T1 加权 MRI 扫描
70 : 获取 MRI 体数据(3D)中的第 70 层切片,即 2D 切片图像。
.astype(np.float32) : 转换数据类型为 float32,以便进行计算。
'''
# 4. 显示 MRI 图像
print(mri_img.shape) # (256, 256)
img_size = mri_img.shape[0] # 获取图像的宽度(假设是正方形图像)。
# 5. 绘制 MRI 图像
plt.figure()
plt.imshow(abs(mri_img)) # 显示 MRI 图像,abs() 主要用于防止负像素值影响显示。
plt.title("Original brain image")
plt.show()
(256, 256)
# set up the first shot
rfactor = 4
num_shots = math.ceil(img_size/rfactor)
print("number of shots: {}".format(num_shots))
# define the regularly spaced samples on a single shot
#nsamples = (np.arange(0,img_size) - img_size//2)/(img_size)
num_samples = img_size
num_samples = (num_samples + 1) // 2
print("number of samples: {}".format(num_samples))
num_revolutions = 1
shot = np.arange(0, num_samples, dtype=np.complex_)
radius = shot / num_samples * 1 / (2 * np.pi) * (1 - np.finfo(float).eps)
angle = np.exp(2 * 1j * np.pi * shot / num_samples * num_revolutions)
# first half of the spiral
single_shot = np.multiply(radius, angle)
# add second half of the spiral
#single_shot = np.append(np.flip(single_shot, axis=0), -single_shot[1:])
single_shot = np.append(np.flip(single_shot, axis=0), -single_shot)
print("number of samples per shot: {}".format(np.size(single_shot)))
k_shots = np.array([], dtype = np.complex_)
#for i in vec_shots:
for i in np.arange(0, num_shots):
shot_rotated = single_shot * np.exp(1j * 2 * np.pi * i / (num_shots * 2))
k_shots = np.append(k_shots, shot_rotated)
print(k_shots.shape)
kspace_loc = np.zeros((len(k_shots),2))
kspace_loc[:,0] = k_shots.real
kspace_loc[:,1] = k_shots.imag
#Plot full initialization
kspace = plt.figure(figsize = (8,8))
#plot shots
plt.scatter(kspace_loc[::4,0], kspace_loc[::4,1], marker = '.')
plt.title("Spiral undersampling R = %d" %rfactor)
axes = plt.gca()
plt.grid()
number of shots: 64 number of samples: 128 number of samples per shot: 256 (16384,)
print(np.arange(0, num_shots))
[ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63]
import warnings
warnings.filterwarnings("ignore") # 忽略所有警告
#data=convert_locations_to_mask(kspace_loc, mri_img.shape)
NufftOperator = mrinufft.get_operator("finufft")
# For better image quality we use a density compensation
density = "voronoi"
#density = "cell_count"
#ensity = None
# And create the associated operator.
if density == None:
nufft = NufftOperator(
kspace_loc.reshape(-1, 2), shape=mri_img.shape, density=None, n_coils=1
)
elif density == "voronoi":
voronoi_weights = get_density("voronoi", kspace_loc.reshape(-1, 2))
nufft = NufftOperator(
kspace_loc.reshape(-1, 2), shape=mri_img.shape, density=voronoi_weights, n_coils=1
)
else:
cell_count_weights = get_density("cell_count", kspace_loc.reshape(-1, 2), shape=mri_img.shape, osf=2.0)
nufft = NufftOperator(
kspace_loc.reshape(-1, 2), shape=mri_img.shape, density=cell_count_weights, n_coils=1, upsampfac=2.0
)
kspace_obs = nufft.op(mri_img) # Image -> Kspace
image2 = nufft.adj_op(kspace_obs) # Kspace -> Image
grid_space = np.linspace(-0.5, 0.5, num=mri_img.shape[0])
grid2D = np.meshgrid(grid_space, grid_space)
#rid_soln = gridded_inverse_fourier_transform_nd(kspace_loc, kspace_obs,
# tuple(grid2D), 'linear')
plt.imshow(np.abs(image2), cmap='gray')
# Calculate SSIM
base_ssim = ssim(image2, mri_img)
plt.title('Gridded Solution\nSSIM = ' + str(base_ssim))
plt.show()
Q1
rfactor (from 4 to 32) in the definition of the number
of shots (i.e. spokes) to perform spiral under-sampling and observe its impact on image
quality (SSIM score).def Reconstruction_v1 (mask, rfactor, density):
'''
mask = "Spiral"
rfactor : from 4 to 32
#density = "voronoi"
#density = "cell_count"
#density = "None"
'''
if mask == "Radial" :
# set up the first shot
## rfactor = 4
nb_shots = math.ceil(img_size/rfactor)
## print("number of shots: {}".format(nb_shots))
# vectorize the nb of shots
vec_shots = np.arange(0,nb_shots)
# define the regularly spaced samples on a single shot
nsamples = (np.arange(0,img_size) - img_size//2)/(img_size)
## print("number of samples per shot: {}".format(np.size(nsamples)))
# NumPy 2.0 : np.complex_ -> np.complex128
shot_c = np.array(nsamples, dtype=np.complex128)
shots = np.array([], dtype=np.complex128)
#shot_c = np.array(nsamples, dtype = np.complex_)
#shots = np.array([], dtype = np.complex_)
# acculumate shots after rotating the initial one by the right angular increment
for k in vec_shots:
shots = np.append(shots, shot_c * np.exp(2 * np.pi * 1j * k/(2*nb_shots)))
kspace_loc = np.zeros((len(shots),2))
#assign real and imaginary parts of complex-valued k-space trajectories to k-space locations
kspace_loc[:,0] = shots.real
kspace_loc[:,1] = shots.imag
'''
#Plot full initialization
kspace = plt.figure(figsize = (8,8))
##plot shots
plt.scatter(kspace_loc[:,0],kspace_loc[:,1], marker = '.')
plt.title("Radial undersampling R = %d" %rfactor)
axes = plt.gca()
plt.grid()
'''
#kspace_loc = mrinufft.initialize_2D_radial(Nc=nb_shots, Ns=nsamples)
## The real deal starts here ##
# Choose your NUFFT backend (installed independly from the package)
NufftOperator = mrinufft.get_operator("finufft")
# For better image quality we use a density compensation
#density = "voronoi"
#density = "cell_count"
#density = None
# And create the associated operator.
if density == None:
nufft = NufftOperator(
kspace_loc.reshape(-1, 2), shape=mri_img.shape, density=None, n_coils=1
)
elif density == "voronoi":
voronoi_weights = get_density("voronoi", kspace_loc.reshape(-1, 2))
nufft = NufftOperator(
kspace_loc.reshape(-1, 2), shape=mri_img.shape, density=voronoi_weights, n_coils=1
)
else:
cell_count_weights = get_density("cell_count", kspace_loc.reshape(-1, 2), shape=mri_img.shape, osf=2.0)
nufft = NufftOperator(
kspace_loc.reshape(-1, 2), shape=mri_img.shape, density=cell_count_weights, n_coils=1, upsampfac=2.0
)
kspace_obs = nufft.op(mri_img) # Image -> Kspace
image2 = nufft.adj_op(kspace_obs) # Kspace -> Image
'''
grid_space = np.linspace(-0.5, 0.5, num=mri_img.shape[0])
grid2D = np.meshgrid(grid_space, grid_space)
#grid_soln = gridded_inverse_fourier_transform_nd(kspace_loc, kspace_obs,
# tuple(grid2D), 'linear')
plt.imshow(np.abs(image2), cmap='gray')
# Calculate SSIM
base_ssim = ssim(image2, mri_img)
plt.title('Adjoint NUFFT solution\nSSIM = ' + str(base_ssim))
plt.show()
'''
# Calculate SSIM
base_ssim = ssim(image2, mri_img)
elif mask == "Spiral" :
# set up the first shot
## rfactor = 4
num_shots = math.ceil(img_size/rfactor)
## print("number of shots: {}".format(num_shots))
# define the regularly spaced samples on a single shot
#nsamples = (np.arange(0,img_size) - img_size//2)/(img_size)
num_samples = img_size
num_samples = (num_samples + 1) // 2
## print("number of samples: {}".format(num_samples))
num_revolutions = 1
shot = np.arange(0, num_samples, dtype=np.complex_)
radius = shot / num_samples * 1 / (2 * np.pi) * (1 - np.finfo(float).eps)
angle = np.exp(2 * 1j * np.pi * shot / num_samples * num_revolutions)
# first half of the spiral
single_shot = np.multiply(radius, angle)
# add second half of the spiral
#single_shot = np.append(np.flip(single_shot, axis=0), -single_shot[1:])
single_shot = np.append(np.flip(single_shot, axis=0), -single_shot)
## print("number of samples per shot: {}".format(np.size(single_shot)))
k_shots = np.array([], dtype = np.complex_)
#for i in vec_shots:
for i in np.arange(0, num_shots):
shot_rotated = single_shot * np.exp(1j * 2 * np.pi * i / (num_shots * 2))
k_shots = np.append(k_shots, shot_rotated)
## print(k_shots.shape)
kspace_loc = np.zeros((len(k_shots),2))
kspace_loc[:,0] = k_shots.real
kspace_loc[:,1] = k_shots.imag
'''
#Plot full initialization
kspace = plt.figure(figsize = (8,8))
#plot shots
plt.scatter(kspace_loc[::4,0], kspace_loc[::4,1], marker = '.')
plt.title("Spiral undersampling R = %d" %rfactor)
axes = plt.gca()
plt.grid()
'''
#data=convert_locations_to_mask(kspace_loc, mri_img.shape)
NufftOperator = mrinufft.get_operator("finufft")
# For better image quality we use a density compensation
density = "voronoi"
#density = "cell_count"
#ensity = None
# And create the associated operator.
if density == None:
nufft = NufftOperator(
kspace_loc.reshape(-1, 2), shape=mri_img.shape, density=None, n_coils=1
)
elif density == "voronoi":
voronoi_weights = get_density("voronoi", kspace_loc.reshape(-1, 2))
nufft = NufftOperator(
kspace_loc.reshape(-1, 2), shape=mri_img.shape, density=voronoi_weights, n_coils=1
)
else:
cell_count_weights = get_density("cell_count", kspace_loc.reshape(-1, 2), shape=mri_img.shape, osf=2.0)
nufft = NufftOperator(
kspace_loc.reshape(-1, 2), shape=mri_img.shape, density=cell_count_weights, n_coils=1, upsampfac=2.0
)
kspace_obs = nufft.op(mri_img) # Image -> Kspace
image2 = nufft.adj_op(kspace_obs) # Kspace -> Image
'''
grid_space = np.linspace(-0.5, 0.5, num=mri_img.shape[0])
grid2D = np.meshgrid(grid_space, grid_space)
#rid_soln = gridded_inverse_fourier_transform_nd(kspace_loc, kspace_obs,
# tuple(grid2D), 'linear')
plt.imshow(np.abs(image2), cmap='gray')
# Calculate SSIM
base_ssim = ssim(image2, mri_img)
plt.title('Gridded Solution\nSSIM = ' + str(base_ssim))
plt.show()
'''
# Calculate SSIM
base_ssim = ssim(image2, mri_img)
return kspace_loc, image2, base_ssim
$\leadsto$ Here we write a function using a mask based on Spiral Undersampling in k-space and reconstructs the MRI image using the Non-Uniform Fast Fourier Transform (NUFFT). The undersampling is controlled by the undersampling factor rfactor and density compensation method density.
It generates Radial k-space Sampling Locations : It defines total number of radial shots in k-space
nb_shots, also the sample points along a single radial linensamplesnormalized between[-0.5, 0.5]; We generate a single spiral arm : The radius is computed asradius = shot / num_samples * 1 / (2 * np.pi) * (1 - np.finfo(float).eps), The angle is defined asangle = np.exp(2 * 1j * np.pi * shot / num_samples * num_revolutions), We generate the first half of the spiral and mirror it to ensure symmetry; We rotate each radial line by an increment of2 * np.pi / (2 * nb_shots); We store the realkspace_loc[:,0]and imaginarykspace_loc[:,1]components of k-space locations.
$\leadsto$ NUFFT adapts to non-uniformly spaced data by modifying the basis functions :
$$ F_{\text{NUFFT}} = DWF $$
where:
$D$ is the density compensation
$W$ is the interpolation function
$F$ is the standard Fourier transform, or
$$ F(u,v) = \sum_{x,y} f(x,y)e^{-2\pi i (ux+vy)} $$
We choose the NUFFT (Non-Uniform FFT) backend $W$ :
NufftOperator = mrinufft.get_operator("finufft"); We apply density compensation $D$ ("voronoi"or"cell_count") if needed.
We compute non-uniform Fourier transform of the MRI image $W$ :
kspace_obs = nufft.op(mri_img); And we reconstruct the image from the undersampled k-space data $W$ byimage2 = nufft.adj_op(kspace_obs)
mask="Spiral"
vector_rfactor = np.linspace(4, 32, 8)
vector_density = ["voronoi", "cell_count", "None"]
fig, axs = plt.subplots(len(vector_rfactor), 2, figsize=(24, 24))
for i in range(len(vector_rfactor)) :
for j in range(2):
mask = "Spiral"
rfactor = vector_rfactor[i]
density = vector_density[0]
kspace_loc, image2, base_ssim = Reconstruction_v1(mask, rfactor, density)
# Plot full initialization
axs[i, 0].scatter(kspace_loc[:,0],kspace_loc[:,1], marker = '.')
axs[i, 0].set_aspect('equal') # 设置轴比例相等,保持正方形
# 显示重建的图像
axs[i, 1].imshow(np.abs(image2), cmap='Greys_r')
axs[i, 0].set_xlabel(f"R = {rfactor}")
# axs[i, 1].set_xlabel(f"SSIM = {base_ssim}")
axs[i, 1].set_xlabel(f"SSIM = {base_ssim:.2f}") # 保留2位小数
axs[i, j].xaxis.set_label_position('top') # 移动 xlabel 到顶部
axs[i, j].set_ylabel(f"density = {density}")
axs[0, 0].set_title("Spiral undersampling")
axs[0, 1].set_title("Adjoint NUFFT solution")
# Add a main title (suptitle) to the figure
fig.suptitle(f"k-space locations and its Reconstruction Results for Mask: {mask}", fontsize=16)
# Display the figure with adjusted layout
plt.tight_layout(rect=[0, 0, 1, 0.96]) # Adjust layout to fit suptitle
plt.show()
$\leadsto$ In the case where mask = Radial, density = "voronoi", we plot 8 reconstructed images with rfactor varying in vector_rfactor = np.linspace(4, 32, 8) :
Initially, as rfactor increases, the quality of the reconstructed image improves until rfactor ≃ 6, after which the image quality starts to degrade.
To quantitatively evaluate the reconstruction accuracy, we can calculate the Structural Similarity Index between the true image and the reconstructed image.
$\leadsto$ We can compute the Structural Similarity Index to evaluate the quality of reconstruction :
$$ SSIM(x,y) = \frac{(2m_x m_y + C_1)(2\Sigma_{xy} + C_2)}{(m_x^2 + m_y^2 + C_1)(\Sigma_{xx}^2 + \Sigma_{yy}^2 + C_2)} $$
where $m$ represents the mean, and $\Sigma$ denotes the covariance, and $C_1$ and $C_2$ are constants.
The result represents the overall similarity between the two images, ranging from [0, 1], where 1 indicates they are identical, and 0 means there is no similarity at all.
In Python, import the module like :
from modopt.math.metrics import ssim
Then, simply use the following line to compute the SSIM score and map :
base_ssim = ssim(image2, mri_img)
mask="Spiral"
vector_rfactor = np.linspace(4, 32, 29)
vector_density = ["voronoi", "cell_count", "None"]
vector_base_ssim = []
vector_image2 = []
for i in range(len(vector_rfactor)) :
mask = "Spiral"
rfactor = vector_rfactor[i]
density = vector_density[0]
kspace_loc, image2, base_ssim = Reconstruction_v1(mask, rfactor, density)
vector_image2.append(image2)
vector_base_ssim.append(base_ssim)
plt.figure(figsize=(10, 6))
plt.plot(vector_rfactor , vector_base_ssim, linestyle='-', color='red', marker='.', markersize=8, markerfacecolor='red', label='Base SSIM')
plt.xlabel('Rfactor')
plt.ylabel('Base SSIM')
plt.title('Base SSIM vs Rfactor')
plt.grid(True)
plt.legend()
plt.show()
The results show that the maximum Base SSIM occurs at rfactor = 6 . Below, we display the reconstructed images for it :
with density = "voronoi".
# Find the index and value of the ;maximum Frobenius norm.
max_index = np.argmax(vector_base_ssim)
image_corresponding_max_index = vector_image2[max_index]
rfactor_corresponding_max_index = vector_rfactor[max_index]
# print(rfactor_corresponding_max_index)
fig, axes = plt.subplots(1, 2, figsize=(8, 4))
axes[0].imshow(np.abs(image_corresponding_max_index), cmap='Greys_r')
axes[0].set_title(f'Rfactor (for Max Base SSIM) : {rfactor_corresponding_max_index:.2f}')
axes[0].axis('off')
axes[1].imshow(mri_img, cmap='Greys_r')
axes[1].set_title("True image")
axes[1].axis('off')
plt.tight_layout()
plt.show()
from scipy.spatial import Voronoi, voronoi_plot_2d
vor = Voronoi(kspace_loc.reshape(-1, 2))
fig = voronoi_plot_2d(vor)
fig = voronoi_plot_2d(vor, show_vertices=False, line_colors='orange',
line_width=2, line_alpha=0.6, point_size=2)
plt.show()
mri_2D = bwdl.get_mri(4, "T2")[150, ...].astype(np.float32)
print(mri_2D.shape)
plt.figure()
plt.imshow(abs(mri_2D))
plt.show()
(434, 362)
from mrinufft import get_density, get_operator, check_backend
traj = initialize_2D_spiral(256, 256).astype(np.float32)
nufft = get_operator("finufft")(traj, mri_2D.shape, density=False)
kspace = nufft.op(mri_2D)
adjoint = nufft.adj_op(kspace)
fig, axs = plt.subplots(1, 3, figsize=(15, 5))
axs[0].imshow(abs(mri_2D))
display_2D_trajectory(traj, subfigure=axs[1])
axs[2].imshow(abs(adjoint))
voronoi_weights = get_density("voronoi", traj)
nufft_voronoi = get_operator("finufft")(
traj, shape=mri_2D.shape, density=voronoi_weights
)
adjoint_voronoi = nufft_voronoi.adj_op(kspace)
fig, axs = plt.subplots(1, 3, figsize=(15, 5))
axs[0].imshow(abs(mri_2D))
display_2D_trajectory(traj, subfigure=axs[1])
axs[2].imshow(abs(adjoint_voronoi))
flat_traj = traj.reshape(-1, 2)
weights = np.sqrt(np.sum(flat_traj**2, axis=1))
nufft = get_operator("finufft")(traj, shape=mri_2D.shape, density=weights)
adjoint_manual = nufft.adj_op(kspace)
fig, axs = plt.subplots(1, 3, figsize=(15, 5))
axs[0].imshow(abs(mri_2D))
axs[0].set_title("Ground Truth")
axs[1].imshow(abs(adjoint))
axs[1].set_title("no density compensation")
axs[2].imshow(abs(adjoint_manual))
axs[2].set_title("manual density compensation")
Text(0.5, 1.0, 'manual density compensation')
Q2
$\leadsto$ No density compensation mechanism : nufft = get_operator("finufft")(traj, mri_2D.shape, density=False)
TThe reconstructed image exhibits saturation and blurring. The quality of the reconstruction is poor.
$\leadsto$ When we choose Voronoi Diagram Based Density Compensation : voronoi_weights = get_density("voronoi", traj)
The reconstruction quality is significantly improved compared to the case with no density compensation.
$\leadsto$ When we choose manual density compensation flat_traj = traj.reshape(-1, 2) weights = np.sqrt(np.sum(flat_traj**2, axis=1))
The reconstruction quality is enhanced similarly to the Voronoi Diagram Based Density Compensation approach.
Q3
rfactor comment the differences in terms of image
quality between radial and spiral undersampling.vector_mask = ["Radial", "Spiral"]
vector_rfactor = np.linspace(4, 32, 8)
vector_density = ["voronoi", "cell_count", "None"]
fig, axs = plt.subplots(len(vector_rfactor), 4, figsize=(24, 24))
for i in range(len(vector_rfactor)) :
for j in [0,1] :
mask = "Radial"
rfactor = vector_rfactor[i]
density = vector_density[0]
kspace_loc, image2, base_ssim = Reconstruction_v1(mask, rfactor, density)
# Plot full initialization
axs[i, 0].scatter(kspace_loc[:,0],kspace_loc[:,1], marker = '.')
axs[i, 0].set_aspect('equal') # 设置轴比例相等,保持正方形
# 显示重建的图像
axs[i, 1].imshow(np.abs(image2), cmap='Greys_r')
axs[i, 0].set_xlabel(f"R = {rfactor}")
# axs[i, 1].set_xlabel(f"SSIM = {base_ssim}")
axs[i, 1].set_xlabel(f"SSIM = {base_ssim:.2f}") # 保留2位小数
axs[i, j].xaxis.set_label_position('top') # 移动 xlabel 到顶部
axs[i, j].set_ylabel(f"density = {density}")
for j in [2,3] :
mask = "Spiral"
rfactor = vector_rfactor[i]
density = vector_density[0]
kspace_loc, image2, base_ssim = Reconstruction_v1(mask, rfactor, density)
# Plot full initialization
axs[i, 2].scatter(kspace_loc[:,0],kspace_loc[:,1], marker = '.')
axs[i, 2].set_aspect('equal') # 设置轴比例相等,保持正方形
# 显示重建的图像
axs[i, 3].imshow(np.abs(image2), cmap='Greys_r')
axs[i, 2].set_xlabel(f"R = {rfactor}")
# axs[i, 1].set_xlabel(f"SSIM = {base_ssim}")
axs[i, 3].set_xlabel(f"SSIM = {base_ssim:.2f}") # 保留2位小数
axs[i, j].xaxis.set_label_position('top') # 移动 xlabel 到顶部
axs[i, j].set_ylabel(f"density = {density}")
axs[0, 0].set_title("Radial undersampling")
axs[0, 1].set_title("Adjoint NUFFT solution")
axs[0, 2].set_title("Spiral undersampling")
axs[0, 3].set_title("Adjoint NUFFT solution")
# Add a main title (suptitle) to the figure
fig.suptitle(f"k-space locations and its Reconstruction Results for Mask: {vector_mask[0]} and {vector_mask[1]}", fontsize=16)
# Display the figure with adjusted layout
plt.tight_layout(rect=[0, 0, 1, 0.96]) # Adjust layout to fit suptitle
plt.show()
$\leadsto$ For the same undersampling factor R,
Spiral underampling yields a higher SSIM (or the quality of reconstruction is better) compared to Radial undersampling .
This may be due to the shape of the undersampling in k-space: Spiral undersampling provides greater frequency variability, while Radial undersampling consists of straight lines, contains less information, and may include some symmetric components.